In 2010, M. Sakai and the first author showed that the topological complexity of a space $X$ coincides with the fibrewise unpointed L-S category of a pointed fibrewise space $\proj_{1} \colon X \times X \to X$ with the diagonal map $\Delta \colon X \to X \times X$ as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for $S^3/Q_8$ where $Q_8$ is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.
[Alternative abstract Notation]In 2010, M. Sakai and the first author showed that the topological complexity of a space 𝑋 coincides with the fibrewise unpointed L-S category of a pointed fibrewise space pr_1 ∶ 𝑋 × 𝑋 → 𝑋 with the diagonal map Δ ∶ 𝑋 → 𝑋 × 𝑋 as its section. In this paper, we describe our algorithm how to determine the fibrewise L-S category or the Topological Complexity of a topological spherical space form. Especially, for 𝑆^3∕_𝑄8 where 𝑄_8 is the quaternion group, we write a python code to realise the algorithm to determine its Topological Complexity.